{"id":11120,"date":"2023-12-12T22:21:21","date_gmt":"2023-12-13T01:21:21","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=11120"},"modified":"2023-12-13T10:44:31","modified_gmt":"2023-12-13T13:44:31","slug":"block-wise-construction-of-bimagic-squares-multiples-of-orders-49-and-343","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=11120","title":{"rendered":"Block-Wise Construction of Bimagic Squares of Orders 49 and 343"},"content":{"rendered":"\n<p class=\"has-text-align-right\"><strong><em>Whole the work is done manually on excel sheets<\/em><\/strong>.<\/p>\n\n\n\n<p class=\"has-text-align-justify has-electric-grass-gradient-background has-background\">Below are <strong>bimagic squares<\/strong> written in blocks multiples of orders 7. These are orders 49 and 343. The work is done manually by author in 2011.  It is summarized in the following link. <br><br><strong>Inder J. Taneja<\/strong>, Bimagic Squares of Bimagic Squares and an Open Problem, Febuarary 11, 2011, 2011,  pp. 1-14, (22.02.2011), h<a href=\"https:\/\/doi.org\/10.48550\/arXiv.1102.3052\">ttps:\/\/doi.org\/10.48550\/arXiv.1102.3052<\/a>.<\/p>\n\n\n\n<p>Before proceeding further below are the basic formulas to check the sums of <strong>magic<\/strong> and <strong>bimagic<\/strong> squares<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Magic Sum<\/strong><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"176\" height=\"88\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/S-2.png\" alt=\"\" class=\"wp-image-11140\"\/><\/figure>\n<\/div>\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Bimagic Sum<\/strong><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"304\" height=\"93\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-2.png\" alt=\"\" class=\"wp-image-11141\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-2.png 304w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-2-300x92.png 300w\" sizes=\"(max-width: 304px) 100vw, 304px\" \/><\/figure>\n<\/div>\n\n\n<p>where <strong><em>n<\/em><\/strong> is the order the magic square.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 49<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>49&#215;49<\/sub>:=58849; Sb<sub>49&#215;49<\/sub>:= 94217249<\/strong>, <\/p>\n\n\n\n<p class=\"has-text-align-justify\">Blocks of order 7 are pandiagonal and of equal sums, <strong>S<sub>7&#215;7<\/sub> :=8407<\/strong>. The magic squares of order 49 is also pandiagonal. See below the <strong>bimagic square<\/strong> of order 49.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"2764\" height=\"1556\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi.png\" alt=\"\" class=\"wp-image-11121\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi.png 2764w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi-300x169.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi-1024x576.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi-768x432.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi-1600x900.png 1600w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi-535x301.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi-1536x865.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi-2048x1153.png 2048w\" sizes=\"(max-width: 2764px) 100vw, 2764px\" \/><\/figure>\n<\/div>\n\n\n<p><strong><em>The work is similar to one studied by Gaston Terry in 1843.<\/em><\/strong><\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 54%,rgb(255,105,0) 100%)\">Excel sheet of Bimagic Square of Order 49<\/h4>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a id=\"wp-block-file--media-66201883-062e-49fd-a7ae-37e8265eab79\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi.xlsx\">49&#215;49-bi<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-66201883-062e-49fd-a7ae-37e8265eab79\">Download<\/a><\/div>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 343: Blocks of Orders 7 and 49<\/h3>\n\n\n\n<p class=\"has-text-align-justify\"><strong>Bimagic Square<\/strong> of order 343 with magic and bimagic sums<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>343&#215;343<\/sub>:=20176975 and Sb<sub>343&#215;343<\/sub>:= 1582540680175<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-justify\">Block of order 49 are <strong>pandiagonal bimagic<\/strong> square with different bimagic sums, <strong>S<sub>49&#215;49<\/sub>:=2882425.<\/strong>  Block of order 7 are also pandiagonal with equal sums <strong>S<sub>7&#215;7<\/sub>:=411775.<\/strong> <\/p>\n\n\n\n<p class=\"has-text-align-justify\">The magic square of order 343 is also pandiagonal. It is tested with software by <a href=\"https:\/\/budshaw.ca\/Download.html\">H. White<\/a>.  <\/p>\n\n\n\n<p><\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 54%,rgb(255,105,0) 100%)\">Excel sheet of Bimagic Square of Order 343<\/h4>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a id=\"wp-block-file--media-2b5cd836-a77c-4ce7-8ee5-69ee2bb1ec4e\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/343x343-bi.xlsx\">343&#215;343-bi<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/343x343-bi.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-2b5cd836-a77c-4ce7-8ee5-69ee2bb1ec4e\">Download<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Whole the work is done manually on excel sheets. Below are bimagic squares written in blocks multiples of orders 7. These are orders 49 and 343. The work is done manually by author in 2011. It is summarized in the following link. Inder J. Taneja, Bimagic Squares of Bimagic Squares and an Open Problem, Febuarary [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":11121,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-11120","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-days-of-year"],"jetpack_featured_media_url":"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11120","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11120"}],"version-history":[{"count":3,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11120\/revisions"}],"predecessor-version":[{"id":11149,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11120\/revisions\/11149"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/11121"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11120"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11120"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11120"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}